Splash Screen

Mathematical Practices 7 Look for and make use of structure. Content Standards G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Mathematical Practices 7 Look for and make use of structure. 3 Construct viable arguments and critique the reasoning of others. CCSS

You found measures of interior angles of polygons. Find measures of inscribed angles. Find measures of angles of inscribed polygons. Then/Now

inscribed angle – an angle whose vertex is on the circle and the sides of the angle are chords of the circle (ex: 52) intercepted arc – an arc of a circle having endpoints on the sides of an inscribed angle, and its other points on the interior of the angle (ex: 86) Vocabulary

Concept

Concept

Use Inscribed Angles to Find Measures A. Find mX. Answer: Example 1

A. Find mX. Answer: mX = 43 Use Inscribed Angles to Find Measures Example 1

Use Inscribed Angles to Find Measures = 2(52) or 104 Example 1

Use Inscribed Angles to Find Measures = 2(52) or 104 Example 1

A. Find mC. A. 47 B. 54 C. 94 D. 188 Example 1

A. Find mC. A. 47 B. 54 C. 94 D. 188 Example 1

B. A. 47 B. 64 C. 94 D. 96 Example 1

B. A. 47 B. 64 C. 94 D. 96 Example 1

Concept

R  S R and S both intercept . Use Inscribed Angles to Find Measures ALGEBRA Find mR. R  S R and S both intercept . mR  mS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer: Example 2

R  S R and S both intercept . Use Inscribed Angles to Find Measures ALGEBRA Find mR. R  S R and S both intercept . mR  mS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer: So, mR = 12(5) – 13 or 47. Example 2

ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 Example 2

ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 Example 2

Write a two-column proof. Given: Prove: ΔMNP  ΔLOP Use Inscribed Angles in Proofs Write a two-column proof. Given: Prove: ΔMNP  ΔLOP Proof: Statements Reasons 1. Given LO  MN 2. If minor arcs are congruent, then corresponding chords are congruent. Example 3

3. Definition of intercepted arc M intercepts and L intercepts . Use Inscribed Angles in Proofs Proof: Statements Reasons 3. Definition of intercepted arc M intercepts and L intercepts . M  L 4. Inscribed angles of the same arc are congruent. MPN  OPL 5. Vertical angles are congruent. ΔMNP  ΔLOP 6. AAS Congruence Theorem Example 3

Write a two-column proof. Given: Prove: ΔABE  ΔDCE Select the appropriate reason that goes in the blank to complete the proof below. Proof: Statements Reasons 1. Given AB  DC 2. If minor arcs are congruent, then corresponding chords are congruent. Example 3

3. Definition of intercepted arc D intercepts and A intercepts . Proof: Statements Reasons 3. Definition of intercepted arc D intercepts and A intercepts . D  A 4. Inscribed angles of the same arc are congruent. DEC  BEA 5. Vertical angles are congruent. ΔDCE  ΔABE 6. ____________________ Example 3

A. SSS Congruence Theorem B. AAS Congruence Theorem C. Definition of congruent triangles D. Definition of congruent arcs Example 3

A. SSS Congruence Theorem B. AAS Congruence Theorem C. Definition of congruent triangles D. Definition of congruent arcs Example 3

Concept

ΔABC is a right triangle because C inscribes a semicircle. Find Angle Measures in Inscribed Triangles ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180 Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer: Example 4

ΔABC is a right triangle because C inscribes a semicircle. Find Angle Measures in Inscribed Triangles ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180 Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer: So, mB = 8(10) – 4 or 76. Example 4

ALGEBRA Find mD. A. 8 B. 16 C. 22 D. 28 Example 4

ALGEBRA Find mD. A. 8 B. 16 C. 22 D. 28 Example 4

Concept

Find Angle Measures INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT. Example 5

Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. mS + mV = 180 mU + mT = 180 mS + 90 = 180 (14x) + (8x + 4) = 180 mS = 90 22x + 4 = 180 22x = 176 x = 8 Answer: Example 5

Answer: So, mS = 90 and mT = 8(8) + 4 or 68. Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. mS + mV = 180 mU + mT = 180 mS + 90 = 180 (14x) + (8x + 4) = 180 mS = 90 22x + 4 = 180 22x = 176 x = 8 Answer: So, mS = 90 and mT = 8(8) + 4 or 68. Example 5

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28 Example 5

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28 Example 5

End of the Lesson